Ava solved the compound inequality [tex]\frac{x}{4}+7\ \textless \ -1[/tex] or [tex]2x-1 \geq 9[/tex] for all possible values of [tex]x[/tex]. Which graph represents the solution set?



Answer :

To solve the compound inequality [tex]\(\frac{x}{4} + 7 < -1\)[/tex] or [tex]\(2x - 1 \geq 9\)[/tex], we need to address each part of the inequality separately and then combine the solutions.

Step 1: Solve [tex]\(\frac{x}{4} + 7 < -1\)[/tex]

1.1. Start by isolating [tex]\(x\)[/tex] on one side of the inequality:
[tex]\[ \frac{x}{4} + 7 < -1 \][/tex]

1.2. Subtract 7 from both sides:
[tex]\[ \frac{x}{4} < -1 - 7 \][/tex]

1.3. Simplify the right-hand side:
[tex]\[ \frac{x}{4} < -8 \][/tex]

1.4. Multiply both sides by 4 to solve for [tex]\(x\)[/tex]:
[tex]\[ x < -32 \][/tex]
The solution to the first part is all [tex]\(x\)[/tex] values less than [tex]\(-32\)[/tex].

Step 2: Solve [tex]\(2x - 1 \geq 9\)[/tex]

2.1. Start by isolating [tex]\(x\)[/tex] on one side of the inequality:
[tex]\[ 2x - 1 \geq 9 \][/tex]

2.2. Add 1 to both sides:
[tex]\[ 2x \geq 9 + 1 \][/tex]

2.3. Simplify the right-hand side:
[tex]\[ 2x \geq 10 \][/tex]

2.4. Divide both sides by 2 to solve for [tex]\(x\)[/tex]:
[tex]\[ x \geq 5 \][/tex]
The solution to the second part is all [tex]\(x\)[/tex] values greater than or equal to 5.

Step 3: Combine the Solutions

Since the compound inequality uses "or," we need to find the union of the two solution sets:
- For the first inequality: [tex]\(x < -32\)[/tex]
- For the second inequality: [tex]\(x \geq 5\)[/tex]

The combined solution set includes all values of [tex]\(x\)[/tex] either less than [tex]\(-32\)[/tex] or greater than or equal to 5.

Final Solution:

The solution set can be written as:
[tex]\[ (-\infty, -32) \cup [5, \infty) \][/tex]

This represents all values of [tex]\(x\)[/tex] that solve the given compound inequality.